Brathwaite was unbeaten on 79 as the tourists reached 214 for two at stumps on day three. Shai Hope was not out 21.
After trailing by 386 on the first innings, they still requiring a further 172 runs to make New Zealand bat again.
The target appeared monumental, but Brathwaite has the memory of the West Indies' last tour of New Zealand to know it is not impossible.
In the first Test on the 2013 tour they were 396 in arrears when forced to follow on.
A Marlon Samuels double century took them to 507 in their second innings on the fifth day and New Zealand ran out of time chasing a target of 112 to win.
Brathwaite, who was on that tour but not required for that Test, played confidently against the New Zealand attack.
He featured in a 72-run partnership with Kieran Powell (40) for the first wicket, a 94-run stand with Shimron Hetmyer (66) for the second wicket and has added a further 48 with Hope.
Hetmyer, only 20 and playing just his fourth Test, easily bettered his previous best 25 against Pakistan seven months ago.
Matt Henry, who toiled without success in the West Indies first innings, took both wickets while first innings destroyer Neil Wagner proved expensive with none for 89.
After 19 wickets fell on the first two days of the Test, there were only two on day three as the wicket browned off from the green cover on day one when the West Indies were rolled for 134 in their first innings.
However there were signs in the final hour that the wicket was starting to take turn which would put more focus on spinners for the remainder of the game.
New Zealand resumed the day at 447-9 and added 73 runs before declaring shortly before lunch with Blundell celebrating his call up to the New Zealand Test team with an unbeaten 107 on debut.
He took 10 off the opening over of the day by Miguel Cummins and continued to pile on the pressure for another 21.4 overs while Boult staunchly defied his tail-end status to hold up the other end.
The number 11 batsman faced 60 balls to be not out 18, although he was dropped when Blundell was on 99.
When Blundell was on 82, the West Indies unsuccessfully reviewed a rejected lbw appeal after a Jason Holder ball flicked off the pads for four byes.
But other than that the 27-year-old's 180-ball innings, which included 13 fours and one six, was near chanceless.
He faced nine scoreless deliveries on 99 before an inside edge off Roston Chase went past square leg for two and he then marked the milestone with a six two balls later before New Zealand declared.
First Published: December 3, 2017, 2:12 PM IST